Method and apparatus for detection of a speckle based physically unclonable function

ABSTRACT

An optical arrangement of at least a coherent light source ( 1 ), a strongly scattering object ( 5 ) (the PUF), and a pixe-lated photo-detector ( 6 ), wherein the pixels are comparable in size with the bright and dark patches of the speckle pattern produced by coherent radiation traversing the scattering object ( 5 ). Quantitively, the pixel size should be roughly λ/NA, where λ is the wave-length, and (i) NA=a/z for free-space geometry, with a being the beam radius and z being the distance between the exit surface of the PUF ( 5 ) and the pixelated detector ( 6 ), or (ii) NA is the numerical aperture of a lens ( 7 ) in an imaging geometry. In a preferred embodiment of the invention, there are tentative requirements that the pixels should be at least smaller than η max λNA and preferably larger than η max λ/NA, where (in an exemplary embodiment) η max =5 and η min =0.05, say. It will be understood by a person skilled in the art that the present invention is concerned with the optical arrangement of the PUF ( 5 ) and the photo-detector ( 6 ), rather than the photo -detector ( 6 ) per se.

This invention relates to a method and apparatus for detection of aspeckle based physically unclonable function for use in, for example,cryptographic applications.

Information security requires a mechanism that provides significantasymmetry in the effort required to make intended and unintended use ofencoded information. Such protection is growing in importance as anincreasing fraction of economic activity is communicated electronically;sending credit card numbers over the Internet or spending money storedin a smart card's memory assumes that this data cannot be easilyduplicated.

Modem cryptographic practice rests on the use of one-way functions.These are functions that are easy to evaluate in the forward directionbut infeasible to compute in the reverse direction without additionalinformation. Although algorithmic one-way functions are widely used,they are facing a number of challenges, which can be addressed by usingcoherent multiple scattering from inhomogeneous structures rather thannumber theory to implement one-way functions.

Laser speckle fluctuations are a familiar demonstration of thesensitivity of the scattering of coherent radiation to the structure ofinhomogeneous media Because any changes in the microstructure of adisordered medium cause an order unity change in its speckle pattern, adiscretely sampled image of speckle intensity provides a fixed-lengthkey that hashes the specification of the 3D spatial distribution ofscatterers.

A known approach to solving the problems associated with algorithmicone-way functions is the use of physical random functions or physicalunclonable functions (PUF), which are essentially random functions boundto a physical device in such a way that it is computationally andphysically infeasible to predict the output of the function withoutactually evaluating it using the physical device.

Thus, the present invention relates to physically unclonable functions(PUF's) based on speckle patterns. In a known system, an object thatscatters light strongly is illuminated with a coherent light source(e.g. a laser) of wavelength λ. The input beam with beam radius a may bemodified with a spatial light modulator (SLM) which gives the beam acheckerboard light pattern of varying amplitude and/or phase. Thischeckerboard pattern is fully blurred when the beam has traversed thescattering structure. A pixelated detector (for example, CCD or CMOSdevice) is placed a distance z behind the scattering structure andrecords the so-called speckle pattern.

The speckle pattern consists of random bright and dark patches. Thescattering structure is a physical function mapping challenges toresponses. The structure is manufactured in such a way that a duplicatecannot be made, hence the unclonability. For example, the structure maycomprise a layer with randomly dispersed particles of a differentrefractive index than the host medium. In short, the scatteringstructure is an embodiment of a PUF, and it will be appreciated thatPUF's can be applied in cryptographic and cryptologic systems, as setout in detail by Ravikanth Pappu, et al, Physical One-Way Functions,Science 297, p 2026 (2002).

When the pixels of the pixelated detector are much larger than thetypical size of the bright and dark patches of the speckle pattern, theintensity will average out and relevant information will be lost,resulting in deteriorated use of the PUF. On the other hand, when thepixels are much smaller than the typical size of the bright and darkpatches of the speckle pattern, adjacent pixels record essentially thesame information. This redundancy is not a fundamental problem, but ithas the adverse effect of increasing processing requirements.

It is therefore an object of the present invention to provide an opticalmethod and apparatus in which criteria are employed to determine thesize of the pixels of the detector that will give rise to detection ofall relevant bits (i.e. the pixels are small enough) without too muchredundancy (i.e. the pixels are large enough).

SUMMARY OF THE INVENTION

Thus, in accordance with the present invention, there is providedoptical apparatus comprising an optical system for providing a coherentradiation source, a strongly scattering object located in the path ofsaid coherent radiation, and a pixelated photo-detector for detecting aspeckle pattern incident thereon, said speckle pattern being produced bysaid said coherent radiation being incident on said strongly scatteringobject, wherein the size of the pixels of said photo-detector isdetermined by the location thereof in said optical apparatus relative tosaid strongly scattering object and is set at substantially the samesize as that of bright and dark patches present in said speckle patternas determined by λ/NA, where λ is the wavelength of said coherentradiation, and NA is the numerical aperture of said optical system.

Also in accordance with the present invention, there is provided amethod of detecting a speckle pattern, comprising irradiating a stronglyscattering object with coherent radiation and providing a pixelatedphoto-detector for receiving the resultant speckle pattern, wherein thesize of the pixels of said photo-detector is determined by the locationthereof relative to said strongly scattering object and is set atsubstantially the same size as that of bright and dark patches presentin said speckle pattern as determined by λ/NA, where λ is the wavelengthof said coherent radiation, and NA is the numerical aperture of saidoptical system.

In one exemplary embodiment of the present invention, the optical systemcomprises a coherent radiation source for providing a coherent radiationbeam of radius a, the photo-detector being located a distance z fromsaid strongly scattering object, wherein NA=a/z. In an alternativeexemplary embodiment of the present invention, the optical system maycomprise a coherent radiation source for providing a coherent radiationbeam having a radius a, and one or more converging optical elementshaving a focal length f, such as a lens or the like, in the path of thecoherent radiation beam between the strongly scattering object and thephoto-detector, wherein NA=a/f In this case, the one or more opticalelements are located a distance v from the strongly scattering objectand a distance b from the photo-detector, wherein 1/v+1/b=1/f.

In both cases, a spatial light modulator may be provided between thestrongly scattering object and the coherent radiation source. One ormore elements with optical power (that is to say the ability to refractlight), such as a lens or the like, may also be provided in theradiation path between the strongly scattering object and the coherentradiation source. Preferably, the formation of a speckle pattern by thestrongly scattering object is an implementation of a physicallyunclonable function.

Beneficially, the pixels of the photo-detector are at least smaller thanη_(max)λ/NA, where η_(max) is a number in the range 1 to 20, morepreferably 1 to 10, and even more preferably 5 to 10. In one specificembodiment of the invention, η_(min) may be 5. Similarly, the pixels ofthe photo-detector are preferably larger than η_(min)λ/NA, where η_(min)is a number between 0 and 2, more preferably between 0 and 1, and evenmore preferably between 0.05 and 0.5. In one specific embodiment of thepresent invention, η_(min) may be 0.05.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other aspects of the present invention will be apparent from,and elucidated with reference to, the embodiments described herein.

Embodiments of the present invention will now be described by way ofexamples only and with reference to the accompanying drawings, in which:

FIG. 1 is a schematic diagram illustrating optical apparatus accordingto a first exemplary embodiment of the present invention;

FIG. 2 is a graphical representation of the normalised intensityprobability distribution in respect of the apparatus of FIG. 1; and

FIG. 3 is a schematic diagram illustrating optical apparatus accordingto a second exemplary embodiment of the present invention.

DETAILED DESCRIPTION OF THE DRAWINGS

Thus, as stated above, it is an object of the present invention toprovide an optical method and apparatus in which criteria are employedto determine the size of the pixels of the detector that will give riseto detection of all relevant bits (i.e. the pixels are small enough)without too much redundancy (i.e. the pixels are large enough). Inpractice, however, the size of the pixels may be fixed, and it is thedistance between the PUF and the detector which is varied in order tomatch pixel size to speckle patch size.

Thus, the present invention has as its aim to make the pixels comparablein size to the bright and dark patches of the speckle pattern. Thetypical size of these patches has been found to be λz/a, provided that ais much smaller than z, which is likely to be the case, in practice.

The maximum size of the pixels may be estimated using the samplingtheorem of band-limited functions. If z is sufficiently large, the fieldat the detector plane is the Fourier-transform of the field at the exitsurface of the PUF. This regime is known as the far-field or Fraunhoferdiffraction regime. In that case, the highest spatial frequency of thefield at the detector turns out to be a/λz. According to the Nyquistcriterion, it is then necessary to sample the signal with at least twicethis spatial frequency bandwidth, which implies that the pixels shouldbe spaced at a distance λz/2a or less. In practice, pixels have a finitesize and thus do not measure the intensity at a point but insteadmeasure the average intensity over the pixel area

The effect of averaging over the pixel area may be estimated from theprobability distribution of the intensity. According to J. W. Goodman,Statistical Properties of Laser Speckle Patterns (in Laser Speckle andRelated Phenomena, J. C. Dainty ed., Springer-Verlag, 1975), theprobability density to measure a normalised intensity x=I/I₀, where I isthe intensity and I₀ the average intensity is given by:

${P(x)} = {\frac{M}{\Gamma(M)}({Mx})^{M - 1}{\exp\left\lbrack {- {Mx}} \right\rbrack}}$

Where M is approximately related to the pixel area S_(m) by:

$M \approx {1 + \frac{S_{m}}{S_{c}}}$

Where S_(c)≈(λz/a)², the typical area of a speckle patch. Furthermore,Γ(M) is the so-called gamma-function, defined by:

Γ(M) = ∫₀⁰⁰𝕕t t^(M − 1)exp [−t].

The probability function is illustrated in FIG. 2 of the drawings forM=1.0025 (the exponentially decaying function), M=26 (the peakedfunction), and M=1.25 (the intermediate, nearly exponentially decayingfunction). These cases correspond to pixel size of 1/10, 10 and 1 timesthe Nyquist sampling frequency, respectively. Clearly, for small pixelareas, the distribution is relatively smooth, whereas for large pixelareas it is more peaked around the average value, approaching a Gaussiandistribution in the limit of very large pixel area values.

As shown, therefore, the intensity probability distribution is arelatively smooth distribution for small pixel area values and a sharplypeaked distribution for large pixel area values. As the measurement ofintensity is always noisy, it follows that there must be an upper limitfor M above which the width of the peaked distribution isindistinguishable from the noise. Clearly, in order to extract alluseful bits, it is necessary to stay well below this upper limit.

As an example, 5λz/a may be taken as a tentative upper limit for thepixel size. The lower limit below which a decrease in pixel size nolonger increases the amount of useful bits follows from the value for Mbelow which the probability distribution hardly deviates from the valuein the limiting case M=1. In one exemplary embodiment, 0.05λz/a may betaken as a tentative lower limit for the pixel size.

Referring to FIG. 1 of the drawings, an optical arrangement according toa first exemplary embodiment of the present invention is illustrated,which arrangement is referred to herein as ‘free space geometry’. Analternative embodiment is referred to herein as ‘imaging geometry’, inwhich the exit surface of the PUF and the detector are in conjugateplanes of a lens, or more generally, an optical system. In the freespace geometry of FIG. 1, it is possible to define NA=a/z, therebymaking the speckle patch size λ/NA. The arrangement comprises a source 1emitting a coherent light beam 2 of wavelength λ, a lens 3 convergingthe beam 2 to a beam of radius a, an SLM 4 imposing a checkerboardpattern on the beam, a PUF 5, and a pixelated detector 6 placed adistance z behind the PUF.

Referring to FIG. 3 of the drawings, an alternative exemplary embodimentof the present invention is further supplemented with an additional lens7 with focal length f, placed a distance v behind the exit surface ofthe PUF and a distance b in front of the detector, where 1/v+1/b=1/fi.e. the detector is placed in the image plane of the exit surface ofthe PUF. As explained above, in this exemplary embodiment of the presentinvention, the exit surface and the detector are in conjugate planes ofa lens 7, or more generally, an optical system. Conjugate planes meansthat the two planes are the object and the image planes of each other.The typical size of the patches in the speckle pattern is once againλ/NA, where the numerical aperture is defined as NA=a/f where a is thebeam radius at the lens and f is the focal length of the lens (providedthat a is much smaller than f, i.e. it holds only for small NA values).

Other exemplary embodiments of the invention, for example, with anoptical imaging system preceding the PUF, are also conceivable.

Thus, in summary, the present invention provides an optical arrangementof at least a coherent light source, a strongly scattering object (thePUF), and a pixelated photo-detector, wherein the pixels are comparablein size with the bright and dark patches of the speckle pattern.Quantitively, as explained above, the pixel size should be roughly λ/NA,where λ is the wavelength, and (i) NA=a/z for the free-space geometrydescribed above, with a being the beam radius and z being the distancebetween the exit surface of the PUF and the pixelated detector, or (ii)NA is the numerical aperture of the lens 7 in the imaging geometrydescribed above. In a preferred embodiment of the invention, there aretentative requirements that the pixels should be at least smaller thanη_(max)λ/NA and preferably larger than η_(min)λ/NA, where η_(max)=5 andη_(min)=0.05. It will be understood by a person skilled in the art thatthe present invention is concerned with the optical arrangement of thePUF and the photo-detector, rather than the photo-detector per se.

It should be noted that the above-mentioned embodiments illustraterather than limit the invention, and that those skilled in the art willbe capable of designing many alternative embodiments without departingfrom the scope of the invention as defined by the appended claims. Inthe claims, any reference signs placed in parentheses shall not beconstrued as limiting the claims. The word “comprising” and “comprises”,and the like, does not exclude the presence of elements or steps otherthan those listed in any claim or the specification as a whole. Thesingular reference of an element does not exclude the plural referenceof such elements and vice-versa. The invention may be implemented bymeans of hardware comprising several distinct elements, and by means ofa suitably programmed computer. In a device claim enumerating severalmeans, several of these means may be embodied by one and the same itemof hardware. The mere fact that certain measures are recited in mutuallydifferent dependent claims does not indicate that a combination of thesemeasures cannot be used to advantage.

1. Optical apparatus comprising an optical system (6, 7) for providing acoherent radiation beam (2), a strongly scattering object (5) located inthe path of said coherent radiation, and a pixelated photo-detector (6)for detecting a speckle pattern incident thereon, said speckle patternbeing produced by said coherent radiation being incident on saidstrongly scattering object (5), wherein the size of the pixels of saidphoto-detector (6) is determined by the location thereof in said opticalapparatus relative to said strongly scattering object (5) and is set atsubstantially the same size as that of bright and dark patches presentin said speckle pattern as determined by λ/NA, where λ is the wavelengthof said coherent radiation, and NA is the numerical aperture of saidoptical system (6, 7).
 2. Optical apparatus according to claim 1,wherein the optical system comprises a coherent radiation source (1) forproviding a coherent radiation beam (2) of radius a, said photo-detector(6) being located a distance z from said strongly scattering object (5),wherein NA=a/z.
 3. Optical apparatus according to claim 1, wherein theoptical system comprises a coherent radiation source (1) for providing acoherent radiation beam (2) of radius a, and one or more convergingoptical elements (7) having a focal length f, in the path of thecoherent radiation beam (2) between said strongly scattering object (5)and said photo-detector (6), wherein NA=a/f.
 4. Optical apparatusaccording to claim 3, wherein the one or more optical elements (7) arelocated a distance v from the strongly scattering object (5) and adistance b from the photo-detector (6), wherein 1/v+1/b=1/f.
 5. Opticalapparatus according to claim 1, wherein a spatial light modulator (4) isprovided between the coherent radiation source (1) and the stronglyscattering object (5).
 6. Optical apparatus according to claim 1,wherein one or more elements (3) with optical power is provided in theradiation path between the coherent radiation source (1) and thestrongly scattering object (5).
 7. Optical apparatus according to claim1, wherein the formation of a speckle pattern by said stronglyscattering object is an implementation of a physically unclonablefunction.
 8. Optical apparatus according to claim 1, wherein thephoto-detector (6) and the strongly scattering object (5) are locatedrelative to each other such that the pixels of the photo-detector (6)are at least smaller than ηmaxλ/NA, where ηmax is a number in the range1 to
 20. 9. Optical apparatus according to claim 8, wherein ηmax is anumber in the range 1 to
 10. 10. Optical apparatus according to claim 9,wherein ηmax is a number in the range 5 to
 10. 11. Optical apparatusaccording to claim 10, wherein ηmax=5.
 12. Optical apparatus accordingto claim 1, wherein the photo-detector (6) and the strongly scatteringobject (5) are located relative to each other such that the pixels ofthe photo-detector (6) are larger than ηminλ/NA, where ηmin is a numberbetween 0 and
 2. 13. Optical apparatus according to claim 12, whereinηmin is a number between 0 and
 1. 14. Optical apparatus according toclaim 13, wherein ηmin is a number in the range 0.05 and 0.5. 15.Optical apparatus according to claim 14, wherein ηmin=0.05.
 16. A methodof detecting a speckle pattern, comprising irradiating a stronglyscattering object (5) with coherent radiation and providing a pixelatedphoto-detector (6) for receiving the resultant speckle pattern, whereinthe size of the pixels of said photo-detector (6) is determined by thelocation thereof relative to said strongly scattering object (5) and isset at substantially the same size as that of bright and dark patchespresent in said speckle pattern as determined by λ/NA, where λ is thewavelength of said coherent radiation, and NA is the numerical apertureof said optical system.